Symplectic manifolds and their lagrangian submanifolds
Symplectic manifolds and their lagrangian submanifolds
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DOI：
10.1016/0001-8708(71)90020-x
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发表时间：
1971-06
影响因子：
1.7
通讯作者：
A. Weinstein
中科院分区：
文献类型：
--
作者：
A. Weinstein
Darboux的一个经典定理可以表述如下：如果Sz是2n维流形M上的最大秩的闭2-形式，且p是M的任一点，则存在坐标系x1，…，x，，y1，.，yn定义在M中p的邻域U上，使得Sz= a1 ~，A dy，+.+ dx，A dy，在U上。本文的目的是在几个方向上推广Darboux定理，并给出推广的一些应用.推广的第一个方向是所考虑的流形是Banach流形，而不是有限维流形.虽然我们的结果在这方面可能有一些使用在理论力学，主要力量的推广是，它要求我们使用的证明方法，而不是归纳的维数M，这一直是标准的证明技术达布定理。新的证明方法首先由Moser [12]在本文中使用，使我们能够获得关于M的闭子流形而不仅仅是点的邻域中的辛结构的定理。另一个副产品是达布定理的等变版本（推论4.3）。我们的主要结果，定理4.1，当应用于M的所谓的Zagrangian子流形时是最有用的。这些，粗略地说，是极大子流形上的IR拉回到零。本质上，定理6.1说这些子流形没有几何不变量。这一事实反过来又有一个令人惊讶的结果（在第6节末尾讨论过），即辛流形的辛自同构群中的恒等式的邻域可以被无穷小辛自同构的李代数中的零邻域平滑地参数化。第7节专门讨论辛流形的叶理329
A classical theorem of Darboux may be stated as follows: if Sz is a closed 2-form of maximal rank on a 2n-dimensional manifold M and p is any point of M, then there exists a coordinate system x1,..., x,, y1,..., yn defined on a neighborhood U of p in M such that Sz= a!~, A dy,+...+ dx, A dy, on U. The purpose of this paper is to generalize Darboux’s theorem in several directions and to give some applications of the generalizations.The first direction of generalization is that the manifolds considered are Banach, rather than finite-dimensional, manifolds. Although our results in this context may have some use in theoretical mechanics, the main force of this generalization is that it requires us to use a method of proof other than induction on the dimension of M, which has been the standard proof technique for Darboux’s theorem. The new method of proof, first used in this context by Moser [12], enables us to obtain theorems concerning symplectic structures in the neighborhood of a closed submanifold, rather than just a point, of M. Another byproduct is an equivariant version of the Darboux theorem (Corollary 4.3). Our main result, Theorem 4.1, is most useful when applied to the so-called Zagrangian submanifolds of M. These, roughly speaking, are maximal submanifolds on which IR pulls back to zero. Essentially, Theorem 6.1 says that these submanifolds have no geometric invariants. This fact, in turn, has the surprising consequence (discussed at the end of Section 6) that a neighborhood of the identity in the symplectic automorphism group of a symplectic manifold may be smoothly parametrized by a neighborhood of zero in the Lie algebra of infinitesimal symplectic automorphisms. Section 7 is devoted to foliations of symplectic manifolds 329